import numpy as np
from PEPit import PEP
from PEPit.functions import ConvexFunction
from PEPit.functions import ConvexIndicatorFunction
from PEPit.primitive_steps import bregman_gradient_step
[docs]
def wc_no_lips_in_bregman_divergence(L, gamma, n, wrapper="cvxpy", solver=None, verbose=1):
"""
Consider the constrainted composite convex minimization problem
.. math:: F_\\star \\triangleq \\min_x \\{F(x) \\equiv f_1(x) + f_2(x)\\},
where :math:`f_1` is convex and :math:`L`-smooth relatively to :math:`h`,
:math:`h` being closed proper and convex,
and where :math:`f_2` is a closed convex indicator function.
This code computes a worst-case guarantee for the **NoLips** method.
That is, it computes the smallest possible :math:`\\tau(n, L)` such that the guarantee
.. math:: \\min_{t\\leqslant n} D_h(x_{t-1}; x_t) \\leqslant \\tau(n, L) D_h(x_\\star; x_0),
is valid, where :math:`x_n` is the output of the **NoLips** method,
where :math:`x_\\star` is a minimizer of :math:`F`,
and where :math:`D_h` is the Bregman divergence generated by :math:`h`.
In short, for given values of :math:`n` and :math:`L`,
:math:`\\tau(n, L)` is computed as the worst-case value of
:math:`\\min_{t\\leqslant n} D_h(x_{t-1}; x_t)` when :math:`D_h(x_\\star; x_0) \\leqslant 1`.
**Algorithm**: This method (also known as Bregman Gradient, or Mirror descent) can be found in,
e.g., [2, Algorithm 1]. For :math:`t \\in \\{0, \\dots, n-1\\}`,
.. math:: x_{t+1} = \\arg\\min_{u} \\{f_2(u)+\\langle \\nabla f_1(x_t) \\mid u - x_t \\rangle + \\frac{1}{\\gamma} D_h(u; x_t)\\}.
**Theoretical guarantee**:
The **upper** guarantee obtained in [2, Proposition 4] is
.. math:: \\min_{t\\leqslant n} D_h(x_{t-1}; x_t) \\leqslant \\frac{2}{n (n - 1)} D_h(x_\\star; x_0),
for any :math:`\\gamma \\leq \\frac{1}{L}`. It is empirically tight.
**References**:
`[1] H.H. Bauschke, J. Bolte, M. Teboulle (2017).
A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications.
Mathematics of Operations Research, 2017, vol. 42, no 2, p. 330-348.
<https://cmps-people.ok.ubc.ca/bauschke/Research/103.pdf>`_
`[2] R. Dragomir, A. Taylor, A. d’Aspremont, J. Bolte (2021).
Optimal complexity and certification of Bregman first-order methods.
Mathematical Programming, 1-43.
<https://arxiv.org/pdf/1911.08510.pdf>`_
Notes:
Disclaimer: This example requires some experience with PEPit and PEPs ([2], section 4).
Args:
L (float): relative-smoothness parameter.
gamma (float): step-size.
n (int): number of iterations.
wrapper (str): the name of the wrapper to be used.
solver (str): the name of the solver the wrapper should use.
verbose (int): level of information details to print.
- -1: No verbose at all.
- 0: This example's output.
- 1: This example's output + PEPit information.
- 2: This example's output + PEPit information + solver details.
Returns:
pepit_tau (float): worst-case value.
theoretical_tau (float): theoretical value.
Example:
>>> L = 1
>>> gamma = 1 / L
>>> pepit_tau, theoretical_tau = wc_no_lips_in_bregman_divergence(L=L, gamma=gamma, n=10, wrapper="cvxpy", solver=None, verbose=1)
(PEPit) Setting up the problem: size of the Gram matrix: 36x36
(PEPit) Setting up the problem: performance measure is the minimum of 10 element(s)
(PEPit) Setting up the problem: Adding initial conditions and general constraints ...
(PEPit) Setting up the problem: initial conditions and general constraints (1 constraint(s) added)
(PEPit) Setting up the problem: interpolation conditions for 3 function(s)
Function 1 : Adding 132 scalar constraint(s) ...
Function 1 : 132 scalar constraint(s) added
Function 2 : Adding 132 scalar constraint(s) ...
Function 2 : 132 scalar constraint(s) added
Function 3 : Adding 121 scalar constraint(s) ...
Function 3 : 121 scalar constraint(s) added
(PEPit) Setting up the problem: additional constraints for 0 function(s)
(PEPit) Compiling SDP
(PEPit) Calling SDP solver
(PEPit) Solver status: optimal (wrapper:cvxpy, solver: MOSEK); optimal value: 0.022222222222201146
(PEPit) Primal feasibility check:
The solver found a Gram matrix that is positive semi-definite
All the primal scalar constraints are verified up to an error of 2.9698465908722937e-15
(PEPit) Dual feasibility check:
The solver found a residual matrix that is positive semi-definite
All the dual scalar values associated with inequality constraints are nonnegative up to an error of 6.016518693845357e-15
(PEPit) The worst-case guarantee proof is perfectly reconstituted up to an error of 3.0942663016198577e-13
(PEPit) Final upper bound (dual): 0.022222222222206895 and lower bound (primal example): 0.022222222222201146
(PEPit) Duality gap: absolute: 5.748873599387139e-15 and relative: 2.586993119726666e-13
*** Example file: worst-case performance of the NoLips_2 in Bregman divergence ***
PEPit guarantee: min_t Dh(x_(t-1); x_t) <= 0.0222222 Dh(x_*; x_0)
Theoretical guarantee: min_t Dh(x_(t-1); x_t) <= 0.0222222 Dh(x_*; x_0)
"""
# Instantiate PEP
problem = PEP()
# Declare two convex functions and a convex indicator function
d = problem.declare_function(ConvexFunction, reuse_gradient=True)
func1 = problem.declare_function(ConvexFunction, reuse_gradient=True)
h = (d + func1) / L
func2 = problem.declare_function(ConvexIndicatorFunction, D=np.inf)
# Define the function to optimize as the sum of func1 and func2
func = func1 + func2
# Start by defining its unique optimal point xs = x_* and its function value fs = F(x_*)
xs = func.stationary_point()
ghs, hs = h.oracle(xs)
# Then define the starting point x0 of the algorithm and its function value f0
x0 = problem.set_initial_point()
gh0, h0 = h.oracle(x0)
gf0, f0 = func1.oracle(x0)
# Set the initial constraint that is the Bregman distance between x0 and x^*
problem.set_initial_condition(hs - h0 - gh0 * (xs - x0) <= 1)
# Compute n steps of the NoLips starting from x0
x1, x2 = x0, x0
gfx = gf0
ghx = gh0
hx1, hx2 = h0, h0
for i in range(n):
x2, _, _ = bregman_gradient_step(gfx, ghx, func2 + h, gamma)
gfx, _ = func1.oracle(x2)
ghx, hx2 = h.oracle(x2)
Dhx = hx1 - hx2 - ghx * (x1 - x2)
# Update the iterates
x1 = x2
hx1 = hx2
# Set the performance metric to the Bregman distance to the optimum
problem.set_performance_metric(Dhx)
# Solve the PEP
pepit_verbose = max(verbose, 0)
pepit_tau = problem.solve(wrapper=wrapper, solver=solver, verbose=pepit_verbose)
# Compute theoretical guarantee (for comparison)
theoretical_tau = 2 / (n * (n - 1))
# Print conclusion if required
if verbose != -1:
print('*** Example file: worst-case performance of the NoLips_2 in Bregman divergence ***')
print('\tPEPit guarantee:\t min_t Dh(x_(t-1); x_t) <= {:.6} Dh(x_*; x_0)'.format(pepit_tau))
print('\tTheoretical guarantee:\t min_t Dh(x_(t-1); x_t) <= {:.6} Dh(x_*; x_0)'.format(theoretical_tau))
# Return the worst-case guarantee of the evaluated method (and the upper theoretical value)
return pepit_tau, theoretical_tau
if __name__ == "__main__":
L = 1
gamma = 1 / L
pepit_tau, theoretical_tau = wc_no_lips_in_bregman_divergence(L=L, gamma=gamma, n=10,
wrapper="cvxpy", solver=None,
verbose=1)